Equalization and residual self-interference suppression using serial localization with indecision

ABSTRACT

A constellation processing module of a receiver groups points of a constellation associated with a transmitted signal into a plurality of subsets. At least two adjacent ones of the subsets have one or more common constellation points so that the at least two adjacent subsets overlap. The constellation processing module also determines a centroid-based value for each of the subsets of constellation points. A non-final equalization stage localizes a search for a final symbol decision using a set of the centroid-based values as constellation points. An interference suppresser suppresses residual self-interference which arises from using the set of centroid-based values as constellation points to localize the search for the final symbol decision instead of the constellation points used to modulate the transmitted signal. A final equalization stage determines the final symbol decision using a subset of the constellation points used to modulate the transmitted signal.

RELATED APPLICATION

The present application is a continuation-in-part of U.S. patent application Ser. No. 12/549,143, filed Aug. 27, 2009, the content of which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present invention generally relates to equalization, and more particularly relates to equalization based on serial localization with indecision and suppression of the residual self-interference which arises due to the use of serial localization with indecision.

BACKGROUND

MLSE (Maximum Likelihood Sequence Estimation) is a demodulation technique that also suppresses ISI (Inter-Symbol Interference) from a signal which is modulated in accordance with a particular constellation and transmitted over a channel. ISI causes the equalization complexity to increase as a power of the constellation size. Relatively large signal constellations such as 16-, 32- and 64-QAM (Quadrature Amplitude Modulation) have been adopted in EDGE (Enhanced Data Rates for GSM Evolution), HSPA (High Speed Packet Access), LTE (Long Term Evolution) and WiMax (Worldwide Interoperability for Microwave Access). In HSPA, multi-code transmission creates even larger effective constellations. Also, MIMO (Multiple-Input, Multiple-Output) schemes with two or more streams have been adopted in HSPA, LTE and WiMax. MIMO implementations also yield relatively large effective constellations. ISI causes equalization complexity to further increase when any of these techniques occur in combination, e.g. multi-code and MIMO.

In the ISI context, the ideal equalization scheme is MLSE, in the sense of maximizing the probability of correctly detecting the transmitted sequence of symbols, or sequences of symbols in the MIMO case. However, the complexity of MLSE increases substantially as a function of the size of the modulation constellation and/or because of the exponential effects of MIMO or multi-codes to the point where MLSE becomes impractical. Less complex solutions are available such as, DFSE (Decision-Feedback Sequence Estimation), DFE (Decision-Feedback Equalization), etc. Each of these solutions attempts to strike a balance between accuracy and complexity.

For a symbol-spaced channel model with memory M in a MIMO environment, the system model is given by:

r _(k) =H _(M) s _(k−M) + . . . +H ₁ s _(k−1) +H ₀ s _(k) +v _(k)  (1)

Here the element H_(m,i,j) of H_(m) describes the channel from transmit antenna j to receive antenna i at a delay of m symbols. The channel matrices are assumed to be constant over the duration of a burst of data, which will be equalized in one shot. The signal s_(k) has symbol constellation Q of size q. The noise v_(k) is white and Gaussian.

The general ISI scenario includes MIMO. Without much loss of generality, consider the case of a single transmitted signal. For a channel with memory M, MLSE operates on the standard highly regular ISI trellis with q^(M) states, and q^(M+1) branches per stage. The storage complexity of MLSE is roughly driven by the number of states, and the computational complexity by the number of branches. As either M or q grows large, the complexity explodes. Stage k of the trellis describes the progression from state (ŝ_(k−M) . . . ŝ_(k−1)) to state (ŝ_(k−M+1) . . . ŝ_(k)). The branch from (ŝ_(k−M) . . . ŝ_(k−1)) to (ŝ_(k−M+1) . . . ŝ_(k)) represents the symbol ŝ_(k). Note that for the ISI trellis, all branches ending in (ŝ_(k−M+1) . . . ŝ_(k)) share the same symbol. For notational simplicity, the states at each stage are labeled 0 to q^(M)−1. Each index represents a distinct value of (ŝ_(k−M+1) . . . ŝ_(k)). A branch is labeled by its starting and ending state pair (j′, j). For each state j, the fan-in l(j) and the fan-out O(j) are the set of incoming and outgoing branches, respectively. For the ISI trellis, all fan-in and fan-out sets have the same size q.

The branch metric of a branch (j′, j) at step k in the MLSE trellis is given by:

e _(k)(j′,j)=|r _(k) −H _(M) ŝ _(k−M) + . . . +H ₀ ŝ _(k)|²  (2)

Without much loss of generality, the trellis is assumed to start at time 0 in state 0. The state metric computation proceeds forward from there. At time k, the state, or cumulative, metric E_(k)(j) of state j is given in terms of the state metrics at time k−1, and the branch metrics at time k is given by:

$\begin{matrix} {{E_{k}(j)} = {\min\limits_{j^{\prime} \in {I{(j)}}}\left( {{E_{k - 1}\left( j^{\prime} \right)} + {e_{k}\left( {j^{\prime},j} \right)}} \right)}} & (3) \end{matrix}$

In addition, the state in l(j) that achieves the minimum is the so-called predecessor of state j, and denoted π_(k−1)(j). Also, the oldest symbol ŝ_(k−M) in the corresponding M-tuple (ŝ_(k−M), . . . ,ŝ_(k−1)) is the tentative symbol decision looking back from state j at time k. It is possible to trace back a sequence over the different states to time 0, by following the chain π_(k−1)(j), π_(k−2)(π_(k−1)(j)), etc. The corresponding symbols ŝ_(k−M), ŝ_(k−M−1), etc, are the tentative decisions of MLSE looking back from state j at time k. In general, looking back from different states at time k, the decisions tend to agree more the older the symbols. That is, the longer the delay for a decision, the better. Typically, there is a chosen delay D, and the final decision about symbol ŝ_(k−m−D) is made by tracing back from the state (ŝ_(k−M+1) . . . ŝ_(k)) with the smallest state metric. We note again, however, that MLSE has exploding complexity, whether due to the size of the modulation itself, or to the exponential effect of ISI.

Another conventional equalization technique is MSA (Multi-Stage Arbitration). MSA involves sifting through a large set of candidates in multiple stages, where each stage rejects some candidates until a single candidate is left after the final stage. One specific example of MSA is generalized MLSE arbitration where the first stage is a linear equalizer. The second stage implements MLSE based on a sparse irregular trellis over a reduced state space. Iterative Tree Search (ITS) has also been used for performing equalization in MIMO QAM environments. ITS exploits the triangular factorization of the channel. In addition, ITS uses the M-algorithm for reducing the search for the best candidate. ITS breaks down the search further, by dividing the QAM constellation in its four quadrants, and representing each quadrant by its centroid in intermediate computations. The selected quadrant itself is subdivided again into its 4 quadrants, and so on. This results in a quaternary tree search. Other conventional approaches give particular attention to the additional error introduced by the use of centroids instead of true symbols. The error is modeled as Gaussian noise whose variance is determined and incorporated in likelihood computations. However, a tight connection is typically made between the centroid representation and the bit mapping from bits to symbols. That is, if a so-called multi-level bit mapping is employed, then identifying a quadrant is equivalent to making a decision on a certain pair of bits. Such constraints place a restriction on bit mappings, restricting the design of subsets.

SUMMARY

Equalization is performed in a series of stages for suppressing ISI. Each stage attempts to further localize the search for a solution for the benefit of the next stage, based on input from the previous stage. The equalization structure is generally referred to herein as serial localization with indecision (SLI). SLI is a lower complexity alternative to MLSE in the ISI scenario. Viewed in isolation, a given SLI equalization stage can be quite indecisive, but makes progress and avoids an irreversible wrong decision. A given equalization stage localizes the solution by inputting a subset representative of the constellation and outputting a further reduced subset. Each stage makes a choice among candidate reduced subsets. Indecision arises from representing the modulation constellation with overlapping subsets. Indecision is beneficial in a multi-stage SLI structure, because indecision discourages an irreversible bad decision in an early stage.

SLI creates residual self-interference by using a set of centroid-based values as constellation points for demodulation instead of the actual constellation points associated with the transmitted signal. However, the residual self-interference is subjected to the same filtering and channel as the received signal. Thus, the residual self-interference is correlated, and can be modeled as a colored noise and suppressed.

According to an embodiment of a method for equalizing ISI present in a received signal corresponding to a transmitted signal carried over a channel, the method includes grouping points of a constellation used to modulate the transmitted signal into a plurality of subsets, at least two adjacent ones of the subsets having one or more common constellation points so that the at least two adjacent subsets overlap. A centroid-based value is determined for each of the subsets of constellation points. The search for a final symbol decision at a non-final equalization stage of a multi-stage equalizer is localized using a set of the centroid-based values as constellation points. The method also suppresses residual self-interference which arises from using the set of centroid-based values as constellation points to localize the search for the final symbol decision instead of the constellation points used to modulate the transmitted signal. The final symbol decision is determined at a final equalization stage of the multi-stage equalizer using a subset of the constellation points used to modulate the transmitted signal.

Of course, the present invention is not limited to the above features and advantages. Those skilled in the art will recognize additional features and advantages upon reading the following detailed description, and upon viewing the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of an embodiment of a receiver including a multi-stage SLI equalizer including a residual self-interference suppresser and a constellation processing module.

FIG. 2 illustrates a block diagram of an embodiment of a two-stage SLI equalizer including a residual self-interference suppresser.

FIG. 3 illustrates a diagram of an embodiment of overlapping constellation subsets for use by a multi-stage SLI equalizer with residual self-interference suppression.

FIG. 4 illustrates a diagram of an embodiment of overlapping ASK constellation subsets for use by a multi-stage SLI equalizer with residual self-interference suppression.

FIG. 5 illustrates a block diagram of another embodiment of a two-stage SLI equalizer including a residual self-interference suppresser.

FIG. 6 illustrates a block diagram of an embodiment of an i-th stage of a multi-stage SLI equalizer including an MLSE component.

FIG. 7 illustrates a block diagram of an embodiment of an N-stage SLI equalizer with residual self-interference suppression.

FIG. 8 illustrates a diagram of an embodiment of overlapping QAM constellation subsets for use by a multi-stage SLI equalizer with residual self-interference suppression.

FIG. 9 illustrates a diagram of another embodiment of overlapping QAM constellation subsets for use by a multi-stage SLI equalizer with residual self-interference suppression.

FIG. 10 illustrates a block diagram of an embodiment of an N-stage SLI equalization structure including a pre-filter.

FIG. 11 illustrates a block diagram of an embodiment of an i-th stage of a multi-stage SLI equalizer including a DFSE component.

FIG. 12 illustrates a block diagram of an embodiment of an i-th stage of a multi-stage SLI equalizer including an MLSE component and a pre-filter for suppressing residual self-interference.

FIG. 13 illustrates a block diagram of an embodiment of an i-th stage of a multi-stage SLI equalizer including a modified MLSE component that suppresses residual self-interference.

DETAILED DESCRIPTION

FIG. 1 illustrates an embodiment of a wireless transmitter 100 in communication with a wireless receiver 110 over a channel 120. The receiver includes a baseband processor 130 and a constellation processing module 140 and a multi-stage SLI equalizer 150 included in or associated with the baseband processor 130. The constellation processing module 140 groups points of a constellation associated with a transmitted signal into a plurality of subsets, e.g., subsets of ASK constellation points, QAM constellation points, etc. At least two adjacent subsets have one or more common constellation points to ensure overlap. In some embodiments, all adjacent subsets have one or more common constellation points to ensure that all adjacent subsets overlap. In each case, the constellation processing module 140 also determines a centroid-based value for each of the subsets of constellation points and groups the centroid-based values into one or more sets. The values included in each set are centroid-based in that they may be actual centroids, approximations of centroids such as integer values or values quantized to a certain finite precision, the closest constellation point to a centroid, etc. More generally, each subset is assigned a representative, which we call a centroid from here on.

The multi-stage SLI equalizer 150 includes a plurality of stages 152, 154 for suppressing ISI and demodulating a received signal. Each of the equalization stages 152 except for the last stage 154 localizes the search for a final sequence decision using the set of centroid-based values input to or selected by the stage 152 as constellation points. The last equalization stage 154 finalizes the decision using a subset of the initial constellation points. This way, each of the equalization stages 152 except for the last stage 154 further localizes the search for a solution using a set of the centroid-based values as constellation points, reducing the overall complexity of the equalizer. The last stage 154 outputs the final solution based on a subset of the actual constellation. The constellation processing module 140 ensures that at least two adjacent subsets of constellation points overlap to reduce the likelihood of demodulation errors, particularly for the earlier equalization stages as will be described in more detail later herein.

FIG. 2 illustrates an embodiment of a 2-stage SLI equalization structure 200 included in the receiver 110 of FIG. 1 for suppressing ISI and demodulating a received signal r_(k). The received signal r_(k) is carried over the channel 120 and originally modulated at the transmitter 100 using symbol constellation Q. The received signal r_(k) has ISI and is given by equation (1) above. One skilled in the art can readily expand the signal model represented by equation (1) to other scenarios such as MIMO and multi-code transmission. The originally transmitted signal has symbol constellation Q of size q. The constellation processing module 140 of the receiver 110 groups the points of constellation Q into a plurality of subsets in a way that ensures at least two adjacent subsets overlap. The constellation processing module 140 also determines a centroid-based value for each of the subsets of constellation points and generates an alternative constellation Q′ including the centroid-based values, not necessarily belonging to Q, for input to a first stage 210 of the 2-stage SLI equalization structure 200.

The first stage 210 of the SLI structure 200 demodulates the received signal r_(k) and suppresses ISI using the alternative constellation Q′. That is, the first equalization stage 210 uses the centroid-based values included in Q′ as constellation points to perform sequence estimation. Each point in Q′ represents a subset of clustered points in Q. In one embodiment, each centroid-based value included in Q′ is the actual centroid for the points of a particular subset of Q. In another embodiment, the centroids are approximated as integer values. In yet another embodiment, each centroid-based value included in Q′ is the constellation point of Q located closest to the corresponding centroid value. Still other types of values may be used which are derived based on the centroids determined from the different subsets of Q.

The first equalization stage 210 outputs a symbol decision s′_(k) ^([1]), which belongs to Q′. The second equalization stage 220 accepts s′_(k) ^([1]) and uses s′_(k) ^([1]) to choose a localized subset Q″ of Q as its own constellation. The decision s′_(k) ^([1]) output by the first equalization stage 210 can be interpreted to be the representative of Q″ in the first equalization stage 210. The second equalization stage 220 outputs the final symbol decision ŝ_(k), which belongs to Q″. The final symbol decision ŝ_(k) output by the second stage 220 is determined based on the original received signal r_(k) and subset Q″, which is selected based on the localized symbol decision s′_(k) ^([1]) output by the first stage 210. In one embodiment, both equalization stages 210, 220 implement MLSE over their respective alphabets, e.g. as previously described herein. Alternatively, the equalization stages 210, 220 implement other types of equalization schemes such as DFSE, DFE, M-algorithm, tree searching, etc.

In one embodiment, the first equalization stage 210 outputs the complete sequence of symbol decisions s′_(k) ^([1]) in one block, before the second stage 220 begins its operation. In another embodiment, the first equalization stage 210 outputs its symbol decisions s′_(k) ^([1]) sequentially, based on a decision delay D, as discussed earlier. Then the second stage 220 can begin its operation as it sequentially accepts the symbols from the first stage 210.

The performance of the 2-stage SLI structure 200 of FIG. 2 is primarily limited by that of the first equalization stage 210, which in turn is determined by the choice of subsets. Performance suffers when no adjacent subsets overlap. Consider the case of disjoint subsets. ML (Maximum Likelihood) receivers implicitly define a decision region (Voronoi region) around each constellation point, consisting of received values closest to that point than any other. The decision region boundaries are polyhedrons (made up of sections of hyperplanes). Two constellation points x and y are neighbors if their decision regions touch. The common part is a section of the hyperplane P(x,y) that bisects the space according to x and y. In degenerate cases, the common part can become a line or a point. Now consider two adjacent subsets X and Y of Q which are available to the first equalization stage 210 of the two-stage SLI structure 200. Subsets X and Y have centroids c(X) and c(Y), respectively. Consider neighbor pair (x,y), where x belongs to X and y belongs to Y. Suppose x is transmitted, and the first equalization stage 210 makes an error and chooses subset Y instead of subset X. The effective decision boundary of the first equalization stage 210 is the hyperplane P(c(X),c(Y)). In contrast, ML would make the choice based on P(x,y). For the sake of comparison, ML can be thought of as making an effective decision between X and Y. Then the effective decision boundary is made up of the sections P(x,y) of different nearest neighbor pairs (x,y).

FIG. 3 illustrates the effective decision boundary between adjacent subsets X and Y in two dimensions, where the hyperplane becomes a straight line and each constellation point is represented by a circle. In contrast, the decision boundary for ML is a piecewise straight jagged line. The discrepancy between these hypothetical decision boundaries leads to a performance loss in SLI.

Ensuring at least two adjacent subsets have overlap smoothes the decision boundary discrepancy. In particular, in the two stage SLI, including nearest neighbor symbols pairs in the overlap of adjacent subsets of the first demodulation stage means that the first demodulation stage does not have to make a decision about those symbols. That decision will be made in the second stage.

With SLI, the search is further localized from one stage to the next, but the final decision is not made until the last stage. In particular, by making nearest neighbor symbols belong to multiple subsets, a later equalization stage (e.g. second stage 220 in FIG. 2) may recover from an error in an earlier stage (e.g. first stage 210 in FIG. 2). In this context, indecision is beneficial.

FIG. 4 illustrates an exemplary embodiment of an 8-ASK constellation grouped into three subsets. The 8-ASK constellation is given by:

Q={−7,−5,−3,−1,+1,+3,+5,+7}  (4)

The three overlapping subsets shown in FIG. 4 have centroids given by:

Q′={−4,0,+4}  (5)

The overlap means that the second equalization stage 220 of the SLI structure 200 of FIG. 2 can often recover from a bad decision by the first equalization stage 210. The two outer subsets shown in FIG. 4 are offsets of one another, and the offset is equal to the centroid difference. SLI complexity can be further reduced by accounting for the highly structured nature of these subsets. Of course, less structured subsets can also be used with SLI.

FIG. 5 illustrates another embodiment of a 2-stage SLI equalization structure 500 for suppressing ISI and demodulating the received signal r_(k). The 2-stage SLI structure 500 shown in FIG. 5 is similar to the one shown in FIG. 2, except it is tailored to the highly structured nature of the ASK constellation subsets shown in FIG. 4. The constellation input to or selected by the first equalization stage 510 is the set of centroids denoted Q′^([1]). The first equalization stage 510 outputs a decision ŝ′_(k) ^([1]) which is the centroid-based value included in Q′^([1]) that most closely corresponds to the signal r_(k). The first equalization stage 510 also generates a re-modulated signal {circumflex over (r)}′_(k) ^([1]) as a function of ŝ′_(k) ^([1]) and the channel 120 over which the signal is carried as given by:

{circumflex over (r)}′_(k) ^([1]) =H _(M) ŝ′ _(k−M) ^([1]) + . . . +H ₀ ŝ′ _(k) ^([1])  (6)

where H₀ represents the channel 120 with ISI.

The first equalization stage 510 removes the re-modulated signal {circumflex over (r)}′_(k) ^([1]) from r_(k) to generate a modified signal r_(k) ^([1]) for input to the second equalization stage 520 as given by:

r _(k) ^([1]) =r _(k) −{circumflex over (r)}′ _(k) ^([1])  (7)

The modified signal r_(k) ^([1]) is then fed to the second stage 520 instead of the original signal r_(k). The second equalization stage 520 determines the final symbol decision ŝ_(k) by performing sequence estimation on the signal r_(k) ^([1]) output by the first stage 510 using the subset Q′^([2]) of constellation points input to or selected by the last stage 520, generating a localized symbol decision ŝ′_(k) ^([2]) associated with the second stage 520. A summer 530 included in or associated with the second equalization stage 520 sums ŝ′_(k) ^([1]) and ŝ′_(k) ^([2]) to generate the final symbol decision ŝ_(k) as given by:

{circumflex over (s)}_(k) =ŝ′ _(k) ^([1])+{circumflex over (s)}′_(k) ^([2])  (8)

To account for change to the input of the second equalization stage 520, constellation Q′^([2]) is the subset of Q centered so its centroid is equal to 0. With regard to the subset embodiment shown in FIG. 4, Q′^([2]) is the middle subset. The decision ŝ′_(k) ^([2]) output by the second equalization stage 520 is an element of Q′^([2]). Considering again an exemplary 8-ASK constellation embodiment, Q′^([1])={−4,0,+4} and Q′^([2])={−3,−1,+1,+3}. The second subset Q′^([2]) corresponds to the 4-ASK constellation. The 2-stage SLI embodiments described herein can be readily extended to any number of desired stages.

In this case as well, the first stage may output the complete sequence of symbol decisions in one block, the second stage begins its operation. In another embodiment, the first demodulation stage outputs its symbol decisions sequentially, based on a decision delay D, as discussed earlier. Then the second stage can begin its operation as it sequentially accepts the symbols from the first stage.

FIG. 6 illustrates an embodiment of the i-th stage 600 of an N-stage SLI equalizer structure where i<N. In one embodiment, the i-th stage 600 includes an MLSE component 610 which operates over a constellation Q′^([i]) to suppress ISI. Let q′^([i]) denote the size of Q′^([i]). The ISI trellis maintained by the MLSE component derives from Q′^([i]). Particularly, the ISI trellis has (q′^([i])) ^(M) states, and (q′^([i])) ^(M+1) branches per stage. Each state has fan-in and fan-out size q′^([i]). The input to the i-th equalization stage 600 is the modified received signal r_(k) ^([i−1]) output by the immediately preceding stage (not shown in FIG. 6). The constellation Q′^([i]) input to the i-th stage 600 includes a set of centroid-based values determined as previously described herein. The MLSE component 610 outputs symbols ŝ_(k) ^([i]) based on Q′^([i]) and r_(k) ^([i−1]). A re-modulator component 620 of the i-th stage 600 generates a re-modulated signal given by:

{circumflex over (r)}′_(k) ^([i]) =H _(M) ŝ′ _(k−M) ^([i]) + . . . +H ₀ ŝ′ _(k) ^([i])  (9)

A signal subtractor component 630 of the i-th stage 600 subtracts {circumflex over (r)}′_(k) ^([i]) from {circumflex over (r)}′_(k) ^([i−1]) to yield a modified received signal r_(k) ^([i]), which is fed to the next equalization stage (not shown in FIG. 6).

FIG. 7 illustrates an embodiment of an N-stage SLI equalization structure 700. The input to the first stage 710 is the original received signal r_(k) ^([0])=r_(k). For the last stage 730, the constellation Q′^([N]) is a subset of Q. There is no need for a re-modulation block in the last stage 730. A summer component 740 included in or associated with the last equalization stage 730 determines the overall final symbol decision by adding all intermediate symbol decisions as given by:

{circumflex over (s)}_(k) =ŝ′ _(k) ^([1]) + . . . +ŝ′ _(k) ^([N])  (10)

In one embodiment, each intermediary stage 720 of the N-stage SLI equalizer 700 has the same structure as the i-th equalization stage 600 shown in FIG. 6. According to this embodiment, the i-th intermediary stage 720 localizes the search for the final symbol decision ŝ_(k) by demodulating a modified version of the received signal r_(k) ^([i−1]) output by the immediately preceding stage using the set of centroid-based values Q′^([i]) input to or selected by the i-th intermediary stage 720 and outputting a localized symbol decision ŝ′_(k) ^([i]) as described previously herein.

The i-th intermediary stage 720 also generates a re-modulated signal {circumflex over (r)}′_(k) ^([i]) as a function of the channel 120 and the localized symbol decision generated by the stage 720. The re-modulated signal {circumflex over (r)}′_(k) ^([i]) is removed from the modified version of the received signal r_(k) ^([i−1]) the immediately preceding stage, e.g. as shown in FIG. 6, to generate a newly modified version of the received signal r_(k) ^([i]) for input to the stage immediately following the i-th intermediary stage 720. Subtracting {circumflex over (r)}′_(k) ^([i]) removes part of the transmitted signal, which acts as self-interference. This enables later stages to operate with less self-interference. The constellation Q′_(k) ^([i]) input to or selected by each of the i intermediary equalization stages 720 includes centroid-based values, which may or may not belong to Q, whereas the constellation Q′^([N]) input to or selected by the last equalization stage 730 is a subset of Q.

Broadly, there is no restriction on how the overlapping subsets used for SLI equalization are defined. Subset size can vary, the number of available subsets can change from stage to stage, etc. For the case of ASK, overlapping subsets can be defined in a way that yields a nested structure and a three subset representation. Consider the general case of 2^(L) ASK, having the constellation given by:

Q={2^(L)+1, . . . ,−1,+1, . . . ,+2^(L)−1}  (11)

Three overlapping subsets are defined, where the first subset contains the 2^(L−1) negative points. The second includes the 2^(L−1) middle points {−2^(L−1)+1, . . . ,+2^(L−1)−1}, corresponding to 2^(L−1)ASK. The third subset includes the 2^(L−1) positive points. The centroids for each of the three subsets are −2^(L−1), 0 and +2 ^(L−1), respectively. The same technique can be used to generate three overlapping subsets for 2^(L−1)ASK, and so on. An N-stage SLI equalization structure can be designed using these subsets with N≦L. Except for the last stage of the N-stage SLI equalizer, the set of centroids input to or selected by the i-th stage is given by:

Q′^([i])={−2^(N−i,0),+2^(N−i)}  (12)

The last stage of the N-stage SLI equalizer has the constellation of 2^(L−N+1)ASK. In particular, for N=L−1, Q′^([N]) ={−3,−1,+i 3}. If the maximum number of stages N=L is used, then Q′^([N])={−1,+1}. The 8-ASK example described above yields a nested subset construction, with L=3 and N=2 stages.

The SLI embodiments described herein can be readily adapted to other modulation schemes such as QAM. The extension of SLI from ASK to QAM is straightforward. Again, in principle, there is no restriction on how the overlapping subsets are defined. In one embodiment, the nested subset design of 2^(L)-ASK can be generalized to 2^(2L)-QAM. Just as QAM can be viewed as taking the product of two ASK constellations to produce the complex QAM constellation, the product of the ASK subsets can be taken to produce the subsets of QAM.

FIG. 8 illustrates an embodiment of generalizing ASK to 16-QAM. Each QAM constellation point shown in FIG. 8 is represented by an ‘X’. The three subsets for ASK, e.g. as shown in FIG. 4, yield nine subsets for 16-QAM as illustrated by the boxes drawn around different groups of constellation points in FIG. 8. Two or more adjacent subsets have overlapping constellation points. The middle subset coincides with 4-QAM, or QPSK (Quadrature Phase-Shift Keying). The nine centroid values for 16-QAM, which are shown as circles in FIG. 8, are also determined from the respective three centroid values for ASK. ASK can be further generalized to 64-QAM.

The design of overlapping subsets need not be based on the component ASK constellation. FIG. 9 illustrates an embodiment of the 16-QAM constellation where each of the subsets is directly determined from the QAM constellation and not derived from ASK. Each QAM constellation point shown in FIG. 9 is represented by an ‘X’ and the subsets are shown as boxes drawn around different groups of constellation points. Again, at least two adjacent subsets have overlapping constellation points. Each of the SLI equalization embodiments described herein, including subset selection, yield a low complexity alternative to MLSE with good performance. SLI-based equalization provides a distinct complexity advantage over MLSE as the state space grows, either with the constellation size or with the channel memory.

Storage complexity of MLSE is driven by the number of states, and the computational complexity by the number of branches. The states are referred to next as s and the branches as b. For a conventional MLSE structure, an estimate of complexity is given by:

q^(M)s+q^(M+1)b  (13)

The complexity of the SLI equalization structures described herein can be estimated by adding the number of states and the number of branches from the multiple stages. This estimate proves to be over-conservative when memory is re-used because the number of states need no be included in the analysis. Regardless, for N-stage SLI-based equalization, the complexity estimate is given by:

$\begin{matrix} {{\sum\limits_{i = 1}^{N}\; {\left( {q^{\prime}}^{\lbrack i\rbrack} \right)^{M}s}} + {\left( {q^{\prime}}^{\lbrack i\rbrack} \right)^{M + 1}b}} & (14) \end{matrix}$

For 16-QAM, a 2-stage SLI structure may be used based on the ASK-based overlapping subsets derived as previously described herein. Such a 2-stage SLI structure yields q′^([1])=9 and q′^([2])=4. For memory M=1, the complexity estimate is then given by:

(9s+81b)+(4s+16b)=13s+97b  (15)

In contrast, conventional MLSE requires 16s+256b. As M grows, the complexity advantage of SLI-based equalization becomes even more evident.

For 64-QAM, either a 2-stage or a 3-stage SLI equalization structure may be used. The 2-stage SLI structure is based on the 8-ASK example previously described herein, with q′^([1])=9 and q′^([2])=16. For M=1, the complexity estimate for the 64-QAM 2-stage SLI structure is given by:

(9s+81b)+(16s+256b)=25s+337b  (16)

The 3-stage SLI equalization structure uses a 3-level partition for 64-QAM, with q′^([1])=9 and q′^([2])=16. The complexity estimate for the 64-QAM 3-stage SLI structure is given by:

2(9s+81b)+(4s+16b)=22s+178b  (17)

In contrast, conventional MLSE requires 64s+4096b. The complexity advantage of SLI is now evident even for M=1, due to the large constellation size.

The SLI-based equalization structures described herein tend to work more efficiently when the self-interference from lagging channel taps is relatively small. Accordingly, it is useful to combine SLI with a pre-filter, whose job is to push the energy of the channel towards the leading tap(s). Doing so increases the total effective channel memory. However, SLI still provides a large complexity advantage over MLSE, even with the larger memory. A DFSE structure based on SLI, which is discussed later herein, further takes advantage of the pre-filter effect while trimming memory size.

FIG. 10 illustrates an embodiment of an SLI equalizer structure 800 including a pre-filter 810 and an N-stage SLI equalizer 820 of the kind previously described herein. The pre-filter 810 can be designed on the fly, for each realization of the channel response. The effective channel response, as determined by a conventional channel estimator 830, is the convolution of the original channel response and the pre-filter 810, and has a larger memory than the original, but with more of its energy in the leading taps, in particular the first tap. A pre-filter design module 830 determines the pre-filter coefficients so that the pre-filter 810 concentrates the channel energy into the first one or more taps. The filter output r′, along with the effective channel tap information, is input to the N-stage SLI equalizer 820 for performing demodulation and ISI suppression as previously described herein. The pre-filter 810 can be flexibly applied to select stages of the N-stage SLI equalizer 820. That is, some stages, in particular early ones, may be empirically or analytically determined to be more sensitive to self-interference from lagging taps, while other stages, in particular later stages, are not. Flexibly applying the pre-filter 810 to particular stages of the SLI equalizer 820 enables more or less complexity and performance to be designed into the SLI equalizer structure 800.

As described previously herein, individual SLI equalization stages may implement MLSE to perform ISI suppression and demodulation. Alternatively, individual SLI equalization stages can implement DFSE or other types of sequence estimation techniques. DFSE is a very effective approximation to MLSE, offering suitable tradeoff between complexity and performance.

FIG. 11 illustrates an embodiment where the MLSE component 610 included in the i-th SLI equalization stage 600 of FIG. 6 is replaced with a DFSE component 900. The DFSE memory M′<M is typically a design parameter. The DFSE trellis maintained by the DFSE component 900 is the same as the MLSE trellis, with q^(M′) states. Referring to the MLSE branch metric expression (2), the symbols (ŝ_(k−M), . . . ,ŝ_(k−1)) represent the previous state as well as the present symbol ŝ_(k) in order to compute the branch metric. With DFSE, the previous state represents only the M′ most recent symbols (ŝ_(k−M), . . . ,ŝ_(k−1)).

In order to produce the older (M−M′) symbols, tentative symbol decisions are made, e.g. as explained earlier with regard to MLSE. That is, the DFSE component 900 traces back from the truncated state (ŝ_(k−M), . . . ,ŝ_(k−M′−1) by following the chain of predecessor states to produce tentative decisions, denoted as ( s _(k−M), . . . , s _(k−M′−1)). As such, all M symbols are available to compute the branch metric. Since some of the tentative decisions ( s _(k−M), . . . , s _(k−M′−1)) may be erroneous, it is desirable to mitigate their contribution to the branch metric as much as possible. The pre-filter 810 described above is suitable for this purpose.

With regard to the SLI equalization structures previously described herein, in a given stage, MLSE can be replaced with DFSE, with or without a pre-filter. This way, DFSE can be flexibly chosen for certain stages, but not others. For instance, a certain stage may have a relatively large effective constellation, and DFSE would help control the complexity of that stage.

The SLI embodiments previously described herein create residual self-interference by using a set of centroid-based values as constellation points for demodulation instead of the actual constellation points associated with the transmitted signal. Because the residual self-interference passes through the same filters and channel as the received signal, the residual self-interference is correlated, and can thus can be modeled as a colored noise and suppressed by an interference suppresser 1000. In one embodiment, the interference suppresser 1000 is a whitening filter implemented as a pre-filter to the MLSE component 610 or DFSE component 900 in one or more of the non-final SLI stages 210, 510, 600, 710, 720. In another embodiment, the interference suppresser 1000 is a whitening filter implemented as part of the MLSE or DFSE components 610, 900. In either case, receiver performance is further improved by suppressing the residual self-interference which arises from using a set of centroid-based values as constellation points to localize the search for a final symbol decision instead of the constellation points used to modulate the transmitted signal.

In more detail, the first equalization stage 210 shown in FIG. 2 or the first equalization stage 510 shown in FIG. 5 uses a centroid-based constellation Q′^([1]) to effectively model the received signal r_(k) as given by:

r _(k)=(h _(M) ŝ _(k−M) ^([1]) + . . . +h ₀ ŝ _(k) ^([1]))+x _(k)  (18)

The total noise signal x_(k) absorbs the contribution of s_(k) ^([2]), i.e. the source of the residual self-interference. That is, the total noise signal can be expressed as:

$\begin{matrix} \begin{matrix} {x_{k} = {\left( {{h_{M}s_{k - M}^{\lbrack 2\rbrack}} + \ldots + {h_{0}s_{k}^{\lbrack 2\rbrack}}} \right) + v_{k}}} \\ {= {w_{k} + v_{k}}} \end{matrix} & (19) \end{matrix}$

The noise term v_(k) is white and Gaussian. The term w_(k) in equation (19) arises because centroid-based values instead of actual constellation points are used to perform demodulation as previously described herein. The term w_(k) thus represents the residual self-interference signal which adversely affects the performance of the non-final equalization stage 210, 510. The corresponding second equalization stage 220, 520 explicitly estimates ŝ′_(k) ^([2]), but this is of no help to the previous stage.

However, w_(k) is a correlated signal, due to the filtering by the channel response h_(k), as can be seen in equation (19). The interference suppresser 1000 can suppress the residual self-interference signal w_(k) by whitening the interference as described in more detail later herein. The second demodulation stage 220, 520 does not exhibit the same residual interference phenomenon, since it is the last stage in these particular embodiments, and Q′^([2]) consists of true constellation points associated with the transmitted signal.

The notion of residual self-interference extends to a multi-stage SLI structure, e.g. of the kind illustrated in FIGS. 7 and 10. In general, SLI can be naturally extended from two stages to multiple stages as previously described herein. That is, each non-final equalization stage further localizes the search for a final symbol decision based on the set of centroid-based values input to or selected by the stage and the last equalization stages finalizes the symbol decision based on a subset of the actual constellation points used to modulate the transmitted signal of interest. One or more of the non-final equalization stages 710, 720 can include the interference suppresser 1000 for whitening the residual self-interference which arises at the corresponding non-final stage.

Returning to the 2-stage SLI embodiments shown in FIGS. 2 and 5, the interference suppresser 1000 models the total noise signal x_(k), including the residual self-interference w_(k), as a colored noise. Also, s′_(k) ^([2]) can be treated as being uncorrelated in time. Since the constellation Q^([2]) employed by the second stage 220, 520 is a subset of the actual transmitted signal constellation with zero centroid, then s′_(k) ^([2]) has a mean of zero. Thus, the power P₂ of s′_(k) ^([2]) is given by:

$\begin{matrix} {P_{2} = {\frac{1}{{q^{\prime}}^{\lbrack 2\rbrack}}{\sum\limits_{s \in {Q^{\prime}}^{\lbrack 2\rbrack}}\; {s}^{2}}}} & (20) \end{matrix}$

The correlation of w_(k) is then given by:

$\begin{matrix} {{R_{w}()} = {P_{2}{\sum\limits_{k}\; {h_{k}^{*}h_{k + }}}}} & (21) \end{matrix}$

where by definition, h_(k)=0 outside the interval 0≦k≦M. The power spectrum of w_(k) is the Fourier transform of R_(w)(l) and can be expressed as:

S_(W)(f)=P₂|H(f)|²  (22)

where H(f) is the Fourier transform of h_(k). The total noise signal x_(k)=w_(k)+v_(k) has correlation given by:

R _(x)(l)=R _(W(l)+) R _(V)(l)  (23)

The power spectrum of x_(k) can thus be expressed as:

S _(x)(f)=P ₂ |H(f)|² +P _(v)  (24)

In one embodiment, the interference suppresser 1000 is a whitening filter designed to transform x_(k) into a white noise. If x_(k) is passed through the whitening filter, {tilde over (x)}_(k) is produced. Let G(f) denote the frequency domain representation of the whitening filter. Then the power spectrum of {tilde over (x)}_(k) can be expressed as given by:

S_({tilde over (x)})(f)=|G(f)|²S_(x)(f)  (25)

Thus, if G(f) is chosen such that:

$\begin{matrix} {{{{G(f)}}^{2} = \frac{1}{S_{x}(f)}},} & (26) \end{matrix}$

then x_(k) can be perfectly whitened.

The choice of G(f) is not unique. There are additional constraints that can be imposed on the design to yield certain desirable features, such as causality. This design problem is well understood in general, and there are numerical methods for generating the whitening filter. There may also be additional constraints for implementation reasons. For instance, it may be desirable to approximate G(f) with an all zero filter having N zeros. The whitening filter can then be implemented in the time domain as g_(k) with (N+1) taps. For ease of explanation only, the whitening filter is presumed to be g_(k) with (N+1) taps. However, other types of whitening filters can be readily employed which adequately suppress the residual self-interference w_(k). Whitening the total noise signal, including the residual self-interference, improves SLI performance in at least two ways. First, the power of the noise is reduced. Second, MLSE is designed for white noise and is thus well-suited for SLI when the residual self-interference is suppressed.

FIG. 12 illustrates an embodiment of an i-th stage 1100 of an N-stage SLI equalizer structure where i<N. According to this embodiment, the interference suppresser 1000 is implemented as a whitening pre-filter 1110 before an MLSE component 1120 of the i-th stage 1100. The signal r_(k) ^([i−1]) input to the i-th stage 1100 is filtered using the pre-filter response g_(k) to produce an alternative input {tilde over (r)}_(k) ^([i−1]) to the MLSE component 1120. The total noise signal x_(k), including the residual self-interference w_(k), is whitened and suppressed by the pre-filter 1110. The effective channel {tilde over (h)}_(k) of the filter output {tilde over (r)}_(k) ^([i−1]) is the convolution of g_(k) and h_(k), and has (M+N+1) taps. The MLSE component 1120 operates on {tilde over (r)}_(k) ^([i−1]) instead of r_(k) ^([i−1]), using the effective channel {tilde over (h)}_(k). As such, the ISI trellis maintained by the MLSE component 1120 has (q′^([1])) ^(M+N) states, and the complexity of the MLSE component 1120 grows accordingly. The symbols ŝ′_(k) ^([i]) out of the MLSE component 1120 are based on Q′^([i]) and {tilde over (r)}_(k) ^([i−1]). A re-modulator component 1130 uses the original channel filter h_(k) to generate a re-modulated signal {circumflex over (r)}′_(k) ^([i]) based on the symbols out of the MLSE component 1120, e.g. as given by equation (9). A signal combiner 1140 subtracts from {circumflex over (r)}′_(k) ^([i]) from r_(k) ^([i−1]) to yield a modified received signal r_(k) ^([i]), which is fed to the next equalization stage (not shown in FIG. 12).

The complexity increase associated with providing the pre-filter 1110 prior to the MLSE component 1120 can be mitigated by using the channel compacting pre-filter 810 previously described herein. According to an embodiment, the pre-filter 1110 is an all-pass filter, which does not affect the noise correlation properties. As a result, the effective channel {tilde over (h)}_(k), when convolved with the channel compacting pre-filter 810, becomes yet another effective channel {tilde over ({tilde over (h)}_(k) with its energy directed toward the leading taps of the channel compacting pre-filter 810. Then DFSE with memory M′<(M+N) can be applied. The number of states (q′^([1])) ^(M′) <(q′^([1])) ^(M+N) , indicating a reduction in complexity. Any loss in performance in comparison to the MLSE embodiment is minimal.

To avoid the equalizer complexity increase associated with the greater number of trellis states which results from using the pre-filter 1110, the original MLSE trellis with (q′^([1])) ^(M) states can be used if the MLSE operation is augmented to include the residual self-interference suppression function.

FIG. 13 illustrates another embodiment of an i-th stage 1200 of an N-stage SLI equalizer structure where i<N. A modified MLSE component 1210 of the i-th stage 1200 suppresses the residual self-interference. In more detail, the modified MLSE component 1210 processes the signal r_(k) ^([i−1]) input from the immediately preceding stage (not shown in FIG. 13). The modified MLSE component 1210 uses the original MLSE trellis with (q′^([1])) ^(M) states, and thus the MSLE state space does not grow according to this embodiment. The modified MLSE component 1210 whitens the total noise signal x_(k), including the residual self-interference w_(k), along the best trellis path leading to a given state. For example, consider branch (j′,j) from state j′ representing ŝ′_(k−1) ^([1])=(ŝ′_(k−M) ^([1]) . . . ŝ′_(k−1) ^([1])) to state j representing ŝ′_(k) ^([1])=(ŝ′_(k−M+1) ^([1]) . . . ŝ′_(k) ^([1])). Branch (j′,j) can be associated with an innovation value given by:

y _(k) =r _(k)−(h _(M) ŝ′ _(k−M) ^([1]) + . . . +h ₀ ŝ′ _(k) ^([1]))  (27)

A corresponding branch metric can be expressed as e_(k)(j′, j)=|y_(k)|². From the viewpoint of the modified MLSE component 1210, the innovation y_(k) is the estimate of the noise at branch (j′,j), assuming symbols (ŝ′_(k−M) ^([1]), . . . ,ŝ′_(k) ^([1])) were transmitted. By comparing equations (27) and (18), it is seen that y_(k) also contains an estimate of the residual interference signal due to s′_(k) ^([2].)

The modified MLSE component 1120 whitens y_(k) based on the pre-filter response g, with (N+1) taps. To do so, the modified MLSE component 1120 also determines the N previous innovation values (y_(k−N), . . . ,y_(k−1)). The modified MLSE component 1120 obtains the N previous innovation values by tracing back N trellis states from j′, and uses the noise values associated with the corresponding branches. The N previous innovation values can be expressed as a whitened innovation given by:

{tilde over (y)}_(k) =g ₀ y _(k) + . . . +g _(N) y _(k−N)  (28)

The MLSE component 1120 modifies the branch metric e_(k)(j′,j) by using the whitened innovation {tilde over (y)}_(k) instead of y_(k), giving:

e_(k)(j′,j)=|{tilde over (y)}_(k)|²  (30)

The modified MLSE component 1120 then computes the corresponding state metric as previously described herein. The symbols {tilde over (s)}′_(k) ^([i]) out of the modified MLSE component 1210 are based on Q′^([i]) and r_(k) ^([i−1]). A re-modulator component 1220 of the i-th stage 1200 uses the original channel filter h_(k) to generate a re-modulated signal {circumflex over (r)}_(k) ^([i]) based on the symbols out of the MLSE component 1120, e.g. as given by equation (9). A signal combiner 1230 of the i-th stage 1200 subtracts {circumflex over (r)}′_(k) ^([i−1]) from r_(k) ^([i−1]) to yield a modified received signal r_(k) ^([i]), which is fed to the next equalization stage (not shown in FIG. 13).

The modified MLSE component 1120 can benefit from using the channel compacting filter 810 previously described herein. Particularly, the MLSE component 1120 implements an all-pass filter function which does not affect the noise correlation properties. The channel compacting pre-filter 810 is designed so that its convolution with the original pre-filter response g_(k) produces an effective response {tilde over (g)}_(k) having most of its energy in the leading taps of the channel compacting pre-filter 810. Then if {tilde over (g)}_(k) is used in the modified MLSE 1120 for whitening innovation y_(k), the contribution of the older values of y_(k) is minimized. Since the older values come from tracing back through the trellis, and may be incorrect, minimizing their contribution improves performance.

The pre-filter and modified MLSE embodiments associated with FIGS. 12 and 13 and described above can be readily extended to multi-stage SLI. Except for the final equalization stage, each non-final equalization stage can be modified accordingly. However, the power that enters into the computation of the noise spectrum S_(x)(f) in equation (24) can be accounted for to further improve SLI performance. Recall that for the first equalization stage 210, 510, P₂ is the average power of the symbols in Q′^([2]). For 2-stage SLI, Q′^([2]) is the constellation of the second (final) equalization stage 220, 520. However, Q′^([2]) is also the centered subset corresponding to the centroid equal to zero in Q′^([1]). For multi-stage SLI, the latter definition is adopted for the ith non-final equalization stage. That is, P_(i+1) is the average power of the centered subset corresponding to the centroid equal to zero in Q′^([i]). Thus P_(i+1) changes from stage to stage, leading to a different S_(x)(f), and a different whitening filter. In general, the centered subset does not coincide with Q′^([i+1]). The exception is when i=N−1 for N-stage SLI. Then the centered subset is equal to Q′^([N]), i.e. the subset of actual constellation points used by the final equalization stage to determine the final symbol decision.

Complexity may be reduced by implementing the residual self-interference suppresser 1000 at a subset of the non-final equalization stages of a multi-stage SLI structure. Notably, P_(i+1) decreases further along the SLI structure, as the residual interference becomes smaller. As such, whitening has a greater impact on performance in the earlier SLI stages, and less impact in the later stages. Accordingly, a reduced complexity variant of the pre-filter and modified MLSE embodiments associated with FIGS. 12 and 13 and described above can be provided, where whitening is performed only in one or more of the earlier stages, but not in one or more of the later stages. As such, the interference suppresser 1000 may not be included in all non-final stages of a multi-stage SLI structure in some embodiments, but just in one or more of the earlier stages.

With the above range of variations and applications in mind, it should be understood that the present invention is not limited by the foregoing description, nor is it limited by the accompanying drawings. Instead, the present invention is limited only by the following claims, and their legal equivalents. 

What is claimed is:
 1. A method of equalizing inter-symbol interference (ISI) present in a received signal corresponding to a transmitted signal carried over a channel, comprising: grouping points of a constellation used to modulate the transmitted signal into a plurality of subsets, at least two adjacent ones of the subsets having one or more common constellation points so that the at least two adjacent subsets overlap; determining a centroid-based value for each of the subsets of constellation points; localizing a search for a final symbol decision at a non-final equalization stage of a multi-stage equalizer using a set of the centroid-based values as constellation points; suppressing residual self-interference which arises from using the set of centroid-based values as constellation points to localize the search for the final symbol decision instead of the constellation points used to modulate the transmitted signal; and determining the final symbol decision at a final equalization stage of the multi-stage equalizer using a subset of the constellation points used to modulate the transmitted signal.
 2. The method of claim 1, comprising: modeling a total noise signal, including the residual self-interference, as a colored noise; and suppressing the colored noise with a whitening filter.
 3. The method of claim 2, comprising implementing the whitening filter as a pre-filter to a maximum likelihood sequence estimator of the non-final equalization stage.
 4. The method of claim 3, comprising: filtering a signal input to the non-final equalization stage using the pre-filter to yield a modified signal for input to the maximum likelihood sequence estimator; and processing the modified signal at the maximum likelihood sequence estimator based on an effective channel response determined by convolving a response of the pre-filter and a response of the channel which carried the transmitted signal.
 5. The method of claim 4, comprising: re-modulating symbols out of the maximum likelihood sequence estimator based on the response of the channel which carried the transmitted signal instead of the effective channel response to generate a re-modulated signal; and removing the re-modulated signal from the signal input to the non-final equalization stage to generate a modified signal for input to the next stage of the multi-stage equalizer.
 6. The method of claim 2, comprising implementing the whitening filter as part of a maximum likelihood sequence estimator of the non-final equalization stage.
 7. The method of claim 6, comprising whitening the total noise signal, including the residual self-interference, at the maximum likelihood sequence estimator along a trellis path leading to state j.
 8. The method of claim 7, comprising: computing an innovation value at the maximum likelihood sequence estimator for each of N states along the trellis path, the whitening filter having N+1 taps and each of the innovation values accounting for the residual self-interference at the respective states; and determining a branch metric for a portion of the trellis path from state j−1 to state j based on the innovation values computed for the N states.
 9. The method of claim 6, comprising: re-modulating symbols out of the maximum likelihood sequence estimator based on a response of the channel which carried the transmitted signal to generate a re-modulated signal; and removing the re-modulated signal from a signal input to the non-final equalization stage to generate a modified signal for input to the next stage of the multi-stage equalizer.
 10. The method of claim 2, comprising transforming the total noise signal, including the residual self-interference, into white noise at the whitening filter.
 11. The method of claim 2, comprising implementing the whitening filter as an all-pass pre-filter having an effective channel, which when convolved with a channel compacting pre-filter, yields another effective channel having energy directed toward one or more leading taps of the channel compacting pre-filter.
 12. A receiver, comprising: a constellation processing module operable to group points of a constellation associated with a transmitted signal into a plurality of subsets, at least two adjacent ones of the subsets having one or more common constellation points so that the at least two adjacent subsets overlap, and determine a centroid-based value for each of the subsets of constellation points; a non-final equalization stage of a multi-stage equalizer operable to localize a search for a final symbol decision using a set of the centroid-based values as constellation points; an interference suppresser operable to suppress residual self-interference which arises from using the set of centroid-based values as constellation points to localize the search for the final symbol decision instead of the constellation points used to modulate the transmitted signal; and a final equalization stage of the multi-stage equalizer operable to determine the final symbol decision using a subset of the constellation points used to modulate the transmitted signal.
 13. The receiver of claim 12, wherein the interference suppresser comprises a whitening filter and is operable to model a total noise signal, including the residual self-interference, as a colored noise and suppress the colored noise with the whitening filter.
 14. The receiver of claim 13, wherein the whitening filter is implemented as a pre-filter to a maximum likelihood sequence estimator of the non-final equalization stage.
 15. The receiver of claim 14, wherein the pre-filter is operable to filter a signal input to the non-final equalization stage to yield a modified signal for input to the maximum likelihood sequence estimator, and the maximum likelihood sequence estimator is operable to process the modified signal based on an effective channel response determined by convolving a response of the pre-filter and a response of the channel which carried the transmitted signal.
 16. The receiver of claim 15, wherein the non-final equalization stage is operable to re-modulate symbols out of the maximum likelihood sequence estimator based on the response of the channel which carried the transmitted signal instead of the effective channel response to generate a re-modulated signal and remove the re-modulated signal from the signal input to the non-final equalization stage to generate a modified signal for input to the next stage of the multi-stage equalizer.
 17. The receiver of claim 13, wherein the whitening filter is implemented as part of a maximum likelihood sequence estimator of the non-final equalization stage.
 18. The receiver of claim 17, wherein the maximum likelihood sequence estimator is operable to whiten the total noise signal, including the residual self-interference, along a trellis path leading to state j.
 19. The receiver of claim 18, wherein the whitening filter has N+1 taps, the maximum likelihood sequence estimator is operable to compute an innovation value for each of N states along the trellis path, each of the innovation values accounting for the residual self-interference at the respective states, and the maximum likelihood sequence estimator is further operable to determine a branch metric for a portion of the trellis path from state j−1 to state j based on the innovation values computed for the N states.
 20. The receiver of claim 17, wherein the non-final equalization stage is operable to re-modulate symbols out of the maximum likelihood sequence estimator based on a response of the channel which carried the transmitted signal to generate a re-modulated signal and remove the re-modulated signal from a signal input to the non-final equalization stage to generate a modified signal for input to the next stage of the multi-stage equalizer.
 21. The receiver of claim 13, wherein the whitening filter is operable to transform the total noise signal, including the residual self-interference, into white noise.
 22. The receiver of claim 13, wherein the whitening filter comprises an all-pass pre-filter having an effective channel, which when convolved with a channel compacting pre-filter of the receiver, yields another effective channel having energy directed toward one or more leading taps of the channel compacting pre-filter.
 23. The receiver of claim 13, wherein the multi-stage equalizer comprises M successive stages, and the whitening filter is implemented at each non-final equalization stage of the multi-stage equalizer as a function of the average power of the set of centroid-based values used as constellation points by the next successive non-final equalization stage to localize the search for the final symbol decision, except for the last non-final equalization stage which implements the whitening filter as a function of the average power of the subset of the constellation points used by the final equalization stage to determine the final symbol decision.
 24. The receiver of claim 13, wherein the multi-stage equalizer comprises M successive stages, and the whitening filter is provided for one or more earlier ones of the non-final equalization stages, but not for one or more later ones of the non-final equalization stages. 